𝔥-Relative cohomology on S1|1 and deformations

Abstract

Over the [Formula: see text]-dimensional supercircle [Formula: see text], we investigate the first [Formula: see text]-relative cohomology space associated with the embedding of the Lie superalgebra [Formula: see text] of contact vector fields in the Lie superalgebra [Formula: see text] of superpseudodifferential operators with smooth coefficients, where [Formula: see text] is the orthosymplectic Lie superalgebra. Likewise, we study the same problem for the affine Lie superalgebra [Formula: see text] instead of [Formula: see text]. We classify [Formula: see text]-trivial deformations of the standard embedding of the Lie superalgebra [Formula: see text] into the Lie superalgebra [Formula: see text]. This approach leads to the deformations of the central charge induced on [Formula: see text] by the canonical central extension of [Formula: see text].